不動(dòng)點(diǎn)迭代總結(jié)

1、非線性算子不動(dòng)點(diǎn)理論是非線性泛函分析的重要組成部分，利用迭代算法逼近非線性算 子不動(dòng)點(diǎn)的越來(lái)越廣泛。從具體的空間（如LP空間或IP空間）到抽象空間（如Hilbert空間， Banach空間，賦范線性空間）；從單值映象到集值映象；從一般意義的映象（如非擴(kuò)張映象， 嚴(yán)格偽壓縮映象；強(qiáng)偽壓縮映象等）到漸進(jìn)意義的映象（如漸進(jìn)非擴(kuò)張映象，漸進(jìn)偽壓縮映 象，k-強(qiáng)漸進(jìn)偽壓縮映象等）；從迭代序列的構(gòu)造（如Mann與Ishikawa迭代序列，具誤差 （或混合誤差）Mann與Ishikawa迭代序列，Halpern迭代序列等）到迭代序列的強(qiáng)（弱） 收斂性，穩(wěn)定性。可以說(shuō)成果豐富。迭代序列構(gòu)成了非線性算子不動(dòng)點(diǎn)理

2、論中的重要問(wèn)題。 在不動(dòng)點(diǎn)理論方面，從20世紀(jì)初著名的Banach壓縮映射原理和Browder不動(dòng)點(diǎn)定理問(wèn)世 以來(lái)，特別是近30年來(lái)，由于實(shí)際需要的推動(dòng)和數(shù)學(xué)工作者的不斷努力，這門科學(xué)的理論 及應(yīng)用的研究已經(jīng)取得重要的進(jìn)展，并且日趨完善。下面我們主要介紹一些近幾年來(lái)不動(dòng)點(diǎn)的迭代格式：首先，我們先看下一算子的發(fā)展一算子1 T稱為非擴(kuò)張的，如果|TX - Ty |x - y|，Vx, y e C。T稱為壓縮的，如果存在a e (0,1)，使得|Tx-Ty|a |x- y|, Vx, y e CT : D (T) ET稱為漸進(jìn)非擴(kuò)張的，如果存在一序列化 e0,8)，limk = 1，使得n n|Tn

3、X-Tny| 1T稱為漸進(jìn)偽壓縮的，如果存在一序列kJ e 0,8),limk廣1，ns，對(duì)任意給定的x, y e D(T)存在j(x y) e J(x y)，使得Tnx 一Tny, j(x 一 y) 1T稱為嚴(yán)格漸進(jìn)偽壓縮的，如果存在一序列kJ e 0,8),limk廣k e (0,1)，n T8，對(duì)任意給定的x, y e D(T)存在j(x y) e J(x y)，使得Tnx Tny, j(x y) 1如果k =1,Vn 1,T稱為偽壓縮的。 nT稱為中間意義的漸進(jìn)非擴(kuò)張的，如果limsup sup (|Tnx一Tny|-1|x- y|) 1，使得lnx -Tn| 1T : K KT稱為強(qiáng)

4、偽壓縮的，如果對(duì)Vx, y e K，存在j(x- y) e J(x- y)和常數(shù)k e (0,1)，滿足 v Tx -Ty, j(x - y) 0, Vx, y e Ev strongly monotone if there exists a positive real number v suchthat Vx - y|2, Vx, y e Efor constant v 0 . This implies that|Bx - By | vx - y|，that is, b is v 一 expansive and when v = 1 , it is expansive.& 一 Lipschi

5、tz continuous if there exists a positive real number& such that|Bx - By | |Bx - By |2, Vx, y e Eclearly, every m 一 cocoercive map a is 1 Lipschitz mcontinuous.Relaxed m 一 cocoercive, if there exists a positive real number m such thatBx - By, x - y -m |Bx - B |2, Vx, y e ERelaxed(m, v) cocoercive , i

6、f there exists a positive realnumber m, v such thatBx - By, x - y -|Bx - B|2 + V|x - y 112, Vx, y e Efor m = 0 , b is v 一 strongly monotone. This class of maps is more general that the class of strongly monotone maps. It is easy to see that we have the following implication: v 一 strongly monotonicit

7、y implying relaxed(m, v) cocoercivity.18平衡問(wèn)題混合平衡問(wèn)題(簡(jiǎn)記為MEP)是求x e C使得下面的不等式成立:F (x,y)+如 x, y - x) 0, y e C.這里F : Cx C R是二元函數(shù)并且中:C H是一個(gè)非線性算子。(MEP)的解集定義為：MEP(F):=f (x,y)+，;甲 x, y x) 0, y e C .如果中=0則成為下面的平衡問(wèn)題，即求xeC使得下面的不等式成立：F(x,y) 0, y e C.它的解集定義為EP(F)。19半群(1)非擴(kuò)張半群，設(shè)G是R +上的一個(gè)非有界集，我們稱序列M := (W ( ) 是C上的一

8、seG個(gè)非擴(kuò)張半群，若它滿足下面的條件：W (0)x = x, Vx e C;W (s +1) = W (s)T (t), Vs, t e G;|W (s)x W (s)y| W (s)x是連續(xù)的。我們用F(W(表示W(wǎng) (s)的不動(dòng)點(diǎn)集，用F(M)表示M的不動(dòng)點(diǎn)集，i.e.,F (W)= c gF (W (s )。(2)Lipschitzian半群，含參量序列:= T (x),0 T 0; 對(duì)每一個(gè)t 0,存在一個(gè)有界可測(cè)函數(shù)L :(0,8)【0,8)使得IT() - (t) t C是連續(xù)的。一個(gè)Lipschitzian半群稱為是非擴(kuò)張的(壓縮的)，若對(duì)所有的t 0，L_t = 1;另外，稱為

9、是漸進(jìn)非擴(kuò)張的，若limsuL) 。1我們用F(T)表示半群r的不動(dòng)點(diǎn)集，i.e., t xT8F(T):= x g C,T(x)=ol其次，我們看一下算法的發(fā)展，二算法Man迭代序列Man迭代序列x (1a +a Tx n+ln n n n修正的Man迭代序列x (1ot+a Tnxn+ln n n n帶平均誤差的修正的Man迭代序列x (1_ot y +a Tnx + y u n+ln n n n n n nIshikawa迭代序列Ishikawa迭代序列fx = (1a+a Ty n+ln n n ny - (1_ 0 )x Txnn n n n修正的Ishikawa迭代序列fx = (

10、1-a+a Tnys n+ln n n ny - (l- P)X + 8 TnX nn n n n 帶平均誤差的修正的Ishikawa迭代序列fx (1_ot y +a Tny + y us n+ln n n n n n ny - (1- p -5 )x + P Tnx +8 vnn n n n n n n3 Happern 迭代x = (1 一以)x +以 Tx4粘性迭代IX0 G Cx =a f (x ) + (1-a )Tx5修正的Reich-Takahashi型迭代序列如果t是具有序列號(hào)u 0,8),k 1的漸進(jìn)非擴(kuò)張映象，則由下式定義的序列x u DIx？x 一 (1 一a y )T

11、jy +a x +y u ,n+1n n +1j0y - (1- P -5 )Tjx + P x +5 v ,+1Ij =0稱為修正的Reich-Takahashi型迭代序列，其中a ,p ,y ,5 是區(qū)間0,1中滿足某些限制的實(shí)數(shù)序列，u ,v 是d中的 有界序列。6漸進(jìn)非擴(kuò)張映射的三步迭代zn 一(1-丫)xn +y ?言/ = (1-P )x +P TnZnn n n nx = (1-a )x +a TnyI n+1n n n n7有限個(gè)非映射族的隱式迭代過(guò)程x = t x + (1-1 )Tx11 01 1 1x = t x + (1-1 )T x22 12 2 2:一、一x = t

12、 x + (1-1 )T xN N N-1N N Nx = t x + (1-1 )T xN+1N+1 NN+1 N +1 N+1:即 x = t x + (1-1 )Tx , T = T,k G I = 1,2,.,Nn n n-1n n n kk (mod N)8雜交迭代(1)非擴(kuò)張映射x0G cy =以 x +(1以)Tx ,n n nn n cn= z G c :| yn - z| | |xn - z|,Q = z g C : 0, TOC o 1-5 h z HYPERLINK l bookmark3 o Current Document nn0 nx = PIn+1CM(2)漸進(jìn)非

13、擴(kuò)張映射x0G cy =以 x + (1以)Tnx , n n nn n* Cn= z G C :| yn - z|2 |xn - z| F+0J,Q = z g C : 0,nn0 nlxn+1 = U其0 =(1-a )(k2 -l)(diamC )2 0,當(dāng) n F9包含平衡問(wèn)題的迭代尤e C,:、，1 ,一一 1;【n+1n nn 1- nn n其中，x , z是由x e C產(chǎn)生的序列，a ,p u(0,1)，尸u【0,2u 。n n1n n=1n n=1n n =110包含半群的迭代x e C,x =(I - a A) j nT (s )x ds + a y f (x ),n三，不動(dòng)

14、點(diǎn)方向的文章1 . G. Cai, C.S. Hu, Strong convergence theorems of modified Ishikawa iterative process with errors for an infinite family of strict pseudo-contractions, Nonlinear Analysis. 71 (2009) 6044-6053. (SCI,二區(qū)，2011IF=1536)C.S. Hu, G. Cai, Viscosity approximation schemes for fixed point problems and e

15、quilibrium problems and variational inequality problems, Nonlinear Analysis. 72 (2010) 1792-1808. (SCI,二區(qū),2011IF=1536)G. Cai, C.S. Hu, On the strong convergence of the implicit iterative processes of a finite family of relatively weak quasi-nonexpansive mappings, Applied Mathematics Letters. 23 (201

16、0) 73-78. (SCI,二區(qū)，2011IF=1.371)G. Cai, C.S. Hu, On strong convergence by the hybrid method for equilibrium and fixed point problems for an infinite family of asymptotically nonexpansive mappings, Fixed Point Theory and Applications. Volume 2009, Article ID 798319, 20 pages, doi:10.1155/2009/798319.

17、(SCI,二區(qū)，2011IF=1643)G. Cai, C.S. Hu, Strong convergence theorems of general iterative process for a finite family of strict pseudo-contractions in q-uniformly smooth Banach spaces, Computers and Mathematics with Applications . 59 (2010) 149-160. (SCI,三 區(qū)，2011IF=1.747)C.S. Hu, G. Cai, Convergence the

18、orems for equilibrium problems and fixed point problems of a finite family of asymptotically k-strictly pseudocontractive mappings in the intermediate sense, Computers and Mathematics with Applications. 61 (2011) 79-93. (SCI,三區(qū)，2011IF=1.747)G. Cai, C.S. Hu, A hybrid approximation method for equilibr

19、ium and fixed problems for a family of infinite nonexpansive mappings and a monotone mapping, Nonlinear Analysis: Hybrid Systems.Shuang Wang, Changsong Hu*, Guoqing Chai, Strong convergence of a new composite iterative method for equilibrium problems and fixed point problems, Applied Mathematics and

20、 Computation, 215 （2010） 3891-3898. （SCI 收錄）8 . Shuang Wang*, A Note on Strong Convergence of a Modified Halperns Iteration for Nonexpansive Mappings, Fixed Point Theory and Applications, 1 （2010） 1-2.（SCI 收錄）Shuang Wang, Changsong Hu*, Guoqing Chai, Hongchang Hu, Equivalent theorems of the converge

21、nce between Ishikawa-Halpern iteration and viscosity approximation method, Applied Mathematics Letters, 23 （2010） 693-699. （SCI 收錄）Shuang Wang*, Changsong Hu, Two new iterative methods for a countable family of nonexpansive mappings in Hilbert spaces, Fixed Point Theory and Applications, 2 （2010） 1-

22、12. （SCI 收錄）Y. L. Song, Strong Convergence Theorems of a New General Iterative Process with Meir-Keeler Contractions for a Countable Family of _i-Strict Pseudocontractions in q-Uniformly Smooth Banach Spaces, Fixed Point Theory and Applications Volume 2010, Article ID 354202, 19 pagesdoi:10.1155/201

23、0/354202.Y. L. Song, A general iteration scheme for variational inequality problem and common fixe d point problems of nonexpansive mappings in q-uniformly smooth Banach spaces, Journal of Global Optimization.Meng Wen, Changsong Hu, Strong convergence of an new iterative method for a zero of accretive operator and nonexpansive mapping, Fixed Point Theory and Applications, 15 June 2012 DOI: 10.1186/1687-181